From [1]}, Lemma 3, for \(d_{\pi ,\gamma }^{P^{*}}(s)\)  we have,
\(\gamma \sum _s {\mathbf {T}}_{\pi }^*(s^{\prime } \,|\,s) d_{\pi ,\gamma }^{P^{*}}(s) - d_{\pi ,\gamma }^{P^{*}}(s^{\prime }) + (1-\gamma ) \mu _0(s^{\prime }) = 0, \quad \forall \, s^{\prime } \in {\mathbb {S}},\) 
